Singular Cardinals and the PCF Theory

1995 ◽  
Vol 1 (4) ◽  
pp. 408-424 ◽  
Author(s):  
Thomas Jech
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Natural Generalization ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Quarter Century ◽  
Pcf Theory ◽  
Cardinal Numbers ◽  
Singular Cardinals ◽  
The Rich

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4. In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory. §2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P(κ), the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ.

Selective families of sets

1980 ◽  
Vol 372 (1750) ◽  
pp. 307-315 ◽  
Keyword(s):  
Cardinal Number ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Choice Functions ◽  
The Given

Consider a cardinal number α, a set I and a family [A v :v in I) of sets. Suppose that for every subset N of I of cardinality less than α we are given a choice of an element x f N v A v for every v in N this paper the author investigates the circumstances under which it is then always possible to make a choice of an element x*of A v for all v in which, in some precisely specified sense, can be approximated arbitrarily closely by some of the given partial choice functions x f . This question has turned out to be important when α is the least infinite cardinal number. Some of the results involve classes of ‘ large ’ cardinals.

On large cardinals and partition relations

1971 ◽  
Vol 36 (2) ◽  
pp. 305-308 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Partition Relation ◽  
Weakly Compact ◽  
Original Result ◽  
Partition Relations ◽  
Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

Inner Models and Large Cardinals

1995 ◽  
Vol 1 (4) ◽  
pp. 393-407 ◽  
Author(s):  
Ronald Jensen
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Ordinal Number ◽  
Infinite Cardinal ◽  
Recursive Definition ◽  
Von Neumann ◽  
Ordinal Numbers ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture: The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: (where } is the power set of x); Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture.

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

2010 ◽  
Vol 3 (1) ◽  
pp. 71-92 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Paraconsistent Logic ◽  
Peano Arithmetic ◽  
Large Cardinals ◽  
Cardinal Numbers ◽  
Naive Set Theory ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
The Continuum

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

A guide to “Coding the universe” by Beller, Jensen, Welch

1985 ◽  
Vol 50 (4) ◽  
pp. 1002-1019 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Affirmative Answer ◽  
Partial Ordering ◽  
Proper Class ◽  
Covering Lemma ◽  
Singular Cardinals ◽  
The Universe

In the wake of Silver's breakthrough on the Singular Cardinals Problem (Silver [74]) followed one of the landmark results in set theory, Jensen's Covering Lemma (Devlin-Jensen [74]): If 0# does not exist then for every uncountable x ⊆ ORD there exists a constructible Y ⊇ X, card(Y) = card(X). Thus it is fair to say that in the absence of large cardinals, V is “close to L”.It is natural to ask, as did Solovay, if we can fairly interpret the phrase “close to L” to mean “generic over L”. For example, if V = L[a], a ⊆ ω and if 0# does not exist then is V-generic over L for some partial ordering ∈ L? Notice that an affirmative answer implies that in the absence of 0#, no real can “code” a proper class of information.Jensen's Coding Theorem provides a negative answer to Solovay's question, in a striking way: Any class can be “coded” by a real without introducing 0#. More precisely, if A ⊆ ORD then there is a forcing definable over 〈L[A], A〉 such that ⊩ V = L[a], a ⊆ ω, A is definable from a. Moreover if 0# ∉ L[A] then ⊩ 0# does not exist. Now as any M ⊨ ZFC can be generically extended to a model of the form L[A] (without introducing 0#) we obtain: For any 〈M, A〉 ⊨ ZFC (that is, M ⊨ ZFC and M obeys Replacement for formulas mentioning A as a predicate) there is an 〈M, A〉-definable forcing such that ⊩ V = L[a], a ⊆ ω, 〈M, A〉 is definable from a. Moreover if 0# ∉ M then ⊩ 0# does not exist.

Continuum cardinals generalized to Boolean algebras

2001 ◽  
Vol 66 (4) ◽  
pp. 1928-1958 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Boolean Algebras ◽  
Infinite Cardinal ◽  
Cardinal Numbers ◽  
Almost Disjoint ◽  
Infinite Sets ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

scholarly journals Borel's conjecture in topological groups

2013 ◽  
Vol 78 (1) ◽  
pp. 168-184 ◽  
Author(s):  
Fred Galvin ◽  
Marion Scheepers
Keyword(s):  
Cardinal Number ◽  
Natural Generalization ◽  
Inaccessible Cardinal ◽  
Proper Class ◽  
Infinite Cardinal ◽  
Topological Groups ◽  
Consistency Results ◽  
Borel Conjecture

AbstractWe introduce a natural generalization of Borel's Conjecture. For each infinite cardinal numberκ, let BCκdenote this generalization. Then BCℕ0is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℕ1is equivalent to the existence of a Kurepa tree of height ℕ1. Using the connection of BCκwith a generalization of Kurepa's Hypothesis, we obtain the following consistency results:(1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℕ1.(2) If it is consistent that BCℕ1, then it is consistent that there is an inaccessible cardinal.(3) If it is consistent that there is a 1-inaccessible cardinal withωinaccessible cardinals above it, then ¬BCℕω+ (∀n<ω)BCℕnis consistent.(4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℕω(5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BCκfor a proper class of cardinalsκof countable cofinality.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

2012 ◽  
Vol 5 (2) ◽  
pp. 269-293 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Paraconsistent Logic ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Cardinal Numbers ◽  
Reflection Theorem ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
Cardinal Arithmetic

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

Reconstruction of the Ancient Numeral System in Basque Language

Philology ◽  
2019 ◽  
Vol 4 (2018) ◽  
pp. 157-172
Author(s):  
FERNANDO GOMEZ-ACEDO ◽  
ENEKO GOMEZ-ACEDO
Keyword(s):  
Cardinal Number ◽  
Counting System ◽  
Cardinal Numbers ◽  
Original Meaning ◽  
Diachronic Development ◽  
Insight Into

Abstract In this work a new insight into the reconstruction of the original forms of the first Basque cardinal numbers is presented and the identified original meaning of the names given to the numbers is shown. The method used is the internal reconstruction, using for the etymologies words that existed and still exist in Basque and other words reconstructed from the proto-Basque. As a result of this work it has been discovered that initially the numbers received their name according to a specific and logic procedure. According to this ancient method of designation, each cardinal number received its name based on the hand sign used to represent it, thus describing the position adopted by the fingers of the hand to represent each number. Finally, the different stages of numerical formation are shown, which demonstrate a long and diachronic development of the whole counting system.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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